The lesson
Read each section in order, work through every example on paper, then use the practice problems and quick check at the bottom.
What makes a relationship proportional?
Two quantities are proportional when their ratio is always the same. On a graph, that means a straight line through the origin (0, 0).
Finding the constant of proportionality
- 1Pick any pair of values (x, y) from a table or graph.
- 2Divide: k = y ÷ x.
- 3Write the equation y = kx.
A table shows (2, 8), (3, 12), (5, 20). Find k and write the equation.
- 1k = y ÷ x = 8 ÷ 2 = 4 (check: 12 ÷ 3 = 4, 20 ÷ 5 = 4).
- 2The equation is y = 4x.
Why this matters
Spot proportional relationships and find the constant of proportionality.
These ideas show up on homework, quizzes, and the next unit in this grade. Getting comfortable here saves time later when the problems stack more steps together.
Common mistakes to avoid
Rushing to the answer without writing steps. Clear work helps you catch mistakes and shows your reasoning.
Re-reading the question too quickly. Underline what you are solving for before you start calculating.
Key ideas from this lesson
- What makes a relationship proportional?
- Finding the constant of proportionality
- Pick any pair of values (x, y) from a table or graph.
- Divide: k = y ÷ x.
- Write the equation y = kx.
Video walkthroughs
Equations of Proportional Relationships
Write y = kx and find the constant of proportionality.
Watch on YouTube
Proportional Relationships
How to tell when two quantities grow together at a steady rate.
Watch on YouTube
Solving Ratio Problems with Tables
Organize equivalent ratios to find the constant k.
Watch on YouTubePractice
For each problem: write your work in the box, type your answer, and check it. If you are stuck, reveal the solution one step at a time. Do not skip straight to the final answer.
Exercise 1
Try it yourselfA table shows x = 3, y = 15. Find the constant of proportionality k.
Step-by-step solution
- 1k = y ÷ x = 15 ÷ 3 = 5.
Exercise 2
Try it yourselfWrite the equation for a proportional relationship with k = 2.5.
Step-by-step solution
- 1For proportional relationships, y = kx.
- 2Substitute k: y = 2.5x.
Exercise 3
Try it yourselfPoints (4, 10) and (6, 15) lie on a line through the origin. Is the relationship proportional? Find k if yes.
Step-by-step solution
- 110 ÷ 4 = 2.5 and 15 ÷ 6 = 2.5 (same ratio).
- 2The line passes through (0,0) when extended, so it is proportional with k = 2.5.
Exercise 4
Try it yourselfy = 3.2x models hours worked (x) and pay (y). How much is earned for 7.5 hours?
Step-by-step solution
- 1Substitute x = 7.5: y = 3.2 × 7.5.
- 23.2 × 7.5 = 24.
- 3Earnings are $24.
Exercise 5
Try it yourselfA graph of a proportional relationship passes through (5, -20). Write the equation.
Step-by-step solution
- 1k = y ÷ x = -20 ÷ 5 = -4.
- 2The equation is y = -4x.
Quick check
Answer all questions. Retake the quiz until you feel confident before moving on.
Proportional Relationships & k
Question 1 of 4
A proportional relationship contains the point (4, 28). What is the constant of proportionality?